NEW YORK (Reuters) - Global equity markets and oil prices bounced back on Tuesday after data showed the vast U.S. services sector extended a three-year expansion in January, while business activity in the euro zone showed signs of recovery.

U.S. and European stocks rallied, with the S&P 500 and Nasdaq gaining more than 1 percent, recouping most of their losses after a sharp sell-off the previous session that was sparked by renewed worries about the euro zone crisis.

A measure of world equity markets also was higher, though only slightly, because of a decline in emerging market shares.

"Yesterday was the first real down day of the year, which shows that we are in this strong bull market. Today we are back to the normal pattern. People are realizing that we've over-reacted to Europe yesterday," said Uri Landesman, president of hedge fund Platinum Partners in New York.

The Institute for Supply Management said its U.S. services sector index eased slightly, to 55.2 last month from 55.7 in December. The reading was in line with economists' forecasts, according to a Reuters survey.

In Europe, Markit's Eurozone Composite PMI, based on business activity across thousands of companies and a good gauge of economic growth, rose in January to a 10-month high of 48.6 from 47.2 the previous month.

The day's data bolstered the view that the world economy was improving, a sentiment that has lifted stock markets around the globe and pushed the benchmark U.S. S&P 500 to a fresh five-year intraday high on Tuesday.

In the biggest leveraged buyout since the financial crisis, Michael Dell reached a deal to take computer maker Dell Inc DELL.O private for $24.4 billion. The move will allow the billionaire chief executive to try to revive the fortunes of his company without Wall Street scrutiny.

Dell shares closed up 1.1 percent at $13.42.

Corporate results also helped the rally. With 56 percent of S&P 500 companies reporting, 68.7 percent posted earnings that beat expectations, or better than the 65 percent rate over the past four quarters or the 62 percent pace since 1994.

MSCI's all-country world equity index .MIWD00000PUS rose 0.28 percent to 354.97, while the FTSEurofirst 300 .FTEU3 index of top European shares closed up 0.3 percent at 1,154.47.

U.S. Treasuries prices fell. The benchmark 10-year U.S. Treasury note was down 17/32 in price to yield 2.016 percent.

Brent crude oil rose 92 cents a barrel to settle at $116.52, while U.S. crude futures settled up 47 cents at $96.64.

"We do not envisage prices receding for any great length of time," said Carsten Fritsch, an analyst at Commerzbank. "The supply-side risks still prevailing, shrinking OPEC supplies and the brightening global economic outlook all suggest that such a retreat is unlikely."

The euro rose against the dollar and yen, returning to its months-long trend of appreciation, as better-than-expected euro zone data affirmed expectations that the European Central Bank will keep policy steady when it meets this week.

The euro, which had taken the brunt of the selling and fallen from a high of over $1.37 at the end of last week to under $1.35 on Monday, rose 0.49 percent to trade at $1.3579.

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